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Extremal Signatures

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Abstract

Let \(E= A-iB\) be a Hermite–Biehler entire function of exponential type \(\tau /2\) where A and B are real entire, and consider \(\mathrm{d}\mu (x) = \mathrm{d}x/|E(x)|^2\). We show that the sign of the product AB is an extremal signature for the space of functions of exponential type \(\tau \) with respect to the norm of \(L^1(\mu )\). This allows us to find best approximations by entire functions of exponential type \(\tau \) in \(L^1(\mu )\)-norm to certain special functions (e.g., the Gaussian and the Poisson kernel).

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Authors and Affiliations

  1. North Dakota State University, 1340 Administration Ave, Fargo, ND, 58105, USA

    Friedrich Littmann

  2. Dakota State University, 820 Washington Ave N, Madison, SD, 57042, USA

    Mark Spanier

Authors
  1. Friedrich Littmann
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  2. Mark Spanier
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Correspondence to Friedrich Littmann.

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Communicated by Edward B. Saff.

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Littmann, F., Spanier, M. Extremal Signatures. Constr Approx 47, 339–356 (2018). https://doi.org/10.1007/s00365-017-9373-7

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  • DOI: https://doi.org/10.1007/s00365-017-9373-7

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